Neither the Arizona Motor Vehicle Division nor the AZDOT Driver's manual note any minimum age to purchase and register a car in the state of Arizona.
However in general a minor cannot enter into a contract so thus cannot be a sole owner of a motor vehicle. A minor or juvenile cannot normally actually own property so his or her parents will normally need to sign on any legal documents surrounding the purchase of a vehicle and thus technically own that property.
There can be exceptions to this generalization depending again upon state laws. For example your home state of Arizona has an emancipation statute for minors over 16 who can show they are sufficiently mature to handle their own financial and health-related affairs. Here a minor to petition the court for an order of emancipation. If the order is granted, the minor would then be able to enter into legally-binding contracts for the purchase of property, including cars.
If you are a high school junior that is not emancipated and still living at home with your parents then whether you buy a car from a dealer or private seller you should need your parent or guardian to sign on any legal documents, such as a purchasing agreement or title. This would also hold true for an auto insurance policy, your parent would need to sign on this legal document as well.
To find out more information about what exactly you will need to do as a minor to purchase, register and insure a motor vehicle in Arizona contact the AZ MVD.